To find the length of angle DC, we first need to find the length of angle DAC.
In triangle BAD, we have the angle BAD = 55 degrees and the angle ABD = 180 - 55 - 32 = 93 degrees (since the sum of angles in a triangle is 180 degrees).
Using the Law of Sines, we can set up the following equation:
(AB / sin(BAD)) = (AD / sin(ABD))
We know that AB = AC = 5 cm and sin(BAD) = sin(55) = 0.8192. Solving for AD, we get:
(5 / 0.8192) = (AD / sin(ABD))
AD = (5 / 0.8192) * sin(ABD)
AD ≈ 6.10 cm
Now, to find angle DAC, we can use the Law of Cosines:
AC^2 = AD^2 + CD^2 - 2 * AD * CD * cos(DAC)
We know that AC = 5 cm and AD = 6.10 cm. Substituting these values, we get:
25 = 6.10^2 + CD^2 - 2 * 6.10 * CD * cos(DAC)
CD^2 - 12.1 * CD * cos(DAC) + 6.10^2 - 25 = 0
CD^2 - 12.1 * CD * cos(DAC) - 10.39 = 0
This is a quadratic equation in terms of CD. Solving for CD using the quadratic formula, we get:
CD = (12.1 * cos(DAC) ± √(12.1^2 * cos(DAC)^2 + 4 * 10.39)) / 2
Since CD is a positive length, we take the positive square root. Substituting cos(DAC) = cos(180 - 55 - 32) = cos(93) ≈ -0.3917, we get:
CD ≈ (12.1 * (-0.3917) + √(12.1^2 * (-0.3917)^2 + 4 * 10.39)) / 2
CD ≈ (-4.749 * 12.1 + √(63.77 + 169.62)) / 2
CD ≈ (-57.6529 + √(233.39)) / 2
CD ≈ (-57.6529 + 15.278) / 2
CD ≈ -42.3749 / 2
CD ≈ -21.1875
Since length cannot be negative, we discard the negative result. Therefore, the length of angle DC is approximately 21.1875 cm.
From a point A on a straight line AB, AC and AC of length 5cm and 7cm are drawing on the same side of AB such that BAC =32 and BAD =55 find the length of angle DC
1 answer
1 answer